Both AL and ALDS rely on the proper regularization parameters \(\lambda _j\)  to avoid from unnecessary bias due to too small \(\lambda _j\)  selection and from large variance due to too large \(\lambda _j\)  choice, so the selection of \(\lambda _j\)  becomes an important task. To tune the \(\lambda _j\) , we follow [1]} who define \(\lambda _j= \lambda |\tilde{\beta }_j|^{-\nu }\) , where \(\lambda \ge 0, \nu >0\)  and \(\tilde{\beta }\)  is the maximum composite likelihood estimate. The weights \(|\tilde{\beta }_j|^{-\nu }\)  serve as a prior knowledge to identify the non-zero coefficients since, for a constant \(\lambda \) , large (resp. small) \(\tilde{\beta }_j\)  will force \(\lambda _j\)  close to zero (resp. infinity) [2]}. The remaining task is to specify \(\lambda \) .
Again, such a data-driven procedure to tune the regularization parameters is standard in the literature. We leave the extension of Theorem REF  with this stochastic choice of \(\lambda _{n,j}\)  for a future work.
